Class 7 Mathematics — Complete Notes with Diagrams

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Chapter 1: Integers

1.1 Number Line Diagram

Negative ←────────────────────────────→ Positive

  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7
  ◄────┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───►
                                │
                              Zero

1.2 Properties of Integer Operations

| Operation | Rule | Example | |-----------|------|---------| | (+) + (+) | Add, keep + | +3 + +5 = +8 | | (−) + (−) | Add, keep − | −3 + (−5) = −8 | | (+) + (−) | Subtract, sign of bigger | +7 + (−3) = +4 | | (+) × (+) | Positive | +4 × +3 = +12 | | (−) × (−) | Positive | −4 × −3 = +12 | | (+) × (−) | Negative | +4 × −3 = −12 |

1.3 Solved Examples

Example 1: Find: (−12) + (−8) + 20

Step 1: Add negative integers: −12 + (−8) = −20
Step 2: Add result to positive: −20 + 20 = 0
Answer: 0

Example 2: Find: (−5) × (−3) × (−2)

Step 1: (−5) × (−3) = +15   [neg × neg = pos]
Step 2: +15 × (−2) = −30    [pos × neg = neg]
Answer: −30

Rule for Odd/Even Number of Negatives:

• Even number of negative integers in product → Positive
• Odd number of negative integers in product → Negative

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Chapter 2: Fractions and Decimals

2.1 Multiplication of Fractions — Diagram

3/4 of 2/3 means:

[████░] × [████░░]
 3 of 4    2 of 3

= (3 × 2)/(4 × 3) = 6/12 = 1/2

Visual: Divide rectangle into 4 cols, 3 rows (12 parts)
Shade 3×2 = 6 parts → 6/12 = 1/2

2.2 Operations Table

| Operation | Method | Example | |-----------|--------|---------| | Multiply fractions | p/q × r/s = pr/qs | 2/3 × 3/5 = 6/15 = 2/5 | | Divide fractions | p/q ÷ r/s = p/q × s/r | 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6 | | Multiply decimals | Count decimal places | 0.3 × 0.4 = 0.12 (2 places) | | Divide decimals | Make divisor whole number | 1.2 ÷ 0.4 = 12 ÷ 4 = 3 |

2.3 Reciprocal Concept

Reciprocal = "flip" the fraction

Number    Reciprocal
  3/4   →   4/3
  5     →   1/5
  1/7   →   7

Rule: a/b × b/a = 1  (product of reciprocals = 1)

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Chapter 3: Data Handling

3.1 Types of Graphs

BAR GRAPH — Comparing categories:

Marks ↑
  90 |    ██
  80 |    ██   ██
  70 | ██  ██   ██   ██
  60 | ██  ██   ██   ██   ██
     └─────────────────────►
       Ali  Bob  Cia  Dev  Eve

Each bar height = value for that category

PIE CHART — Showing parts of a whole:

        Science 30%
       ┌──────────┐
 Maths │ ████████ │ English 25%
  20%  │    ○     │
       │          │
       └──────────┘
        Hindi 25%

Total must = 100%
Each slice angle = (percentage/100) × 360°

3.2 Mean, Median, Mode

Dataset: Marks of 7 students: 45, 62, 75, 45, 88, 62, 45

MEAN   = Sum ÷ Count
       = (45+62+75+45+88+62+45) ÷ 7
       = 422 ÷ 7 = 60.28

MEDIAN = Middle value (arrange in order first)
Arranged: 45, 45, 45, 62, 62, 75, 88
           1   2   3   4   5   6   7
                       ↑
                  Median = 62 (4th value)

MODE   = Most frequent value = 45 (appears 3 times)

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Chapter 4: Simple Equations

4.1 What is an Equation?

EQUATION has two sides separated by = sign:

    Left Side  =  Right Side
       2x + 3  =     11
           ↑              ↑
      Expression      Expression

Goal: Find value of x that makes both sides EQUAL

4.2 Solving Method — Balancing

Equation: 2x + 3 = 11

What we do to LEFT side, we MUST do to RIGHT side:

Step 1: Subtract 3 from both sides
    2x + 3 - 3 = 11 - 3
    2x = 8

Step 2: Divide both sides by 2
    2x ÷ 2 = 8 ÷ 2
    x = 4

CHECK: 2(4) + 3 = 8 + 3 = 11 ✓

4.3 Transposition Method

Transposition: Move terms across = sign, CHANGE the sign

     3x - 5 = 16
     3x = 16 + 5      [−5 moves to right as +5]
     3x = 21
     x = 21/3 = 7

Rule:
  + moves across as −
  − moves across as +
  × moves across as ÷
  ÷ moves across as ×

4.4 Word Problem Examples

Problem: A number when doubled and added to 7 gives 25. Find the number.

Let number = x
2x + 7 = 25
2x = 25 − 7 = 18
x = 9
Answer: The number is 9

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Chapter 5: Lines and Angles

5.1 Types of Angle Pairs — Diagrams

COMPLEMENTARY ANGLES (sum = 90°):

        │
        │  40°
   ─────┼────
        │ 50°
     50° + 40° = 90°

SUPPLEMENTARY ANGLES (sum = 180°):

    120°│ 60°
   ─────┼──────────
     120° + 60° = 180°

VERTICALLY OPPOSITE ANGLES (equal when two lines cross):

       A
       │
   C───┼───D      ∠A = ∠B (vertically opposite, equal)
       │           ∠C = ∠D (vertically opposite, equal)
       B

5.2 Parallel Lines with Transversal

         t (transversal)
         │
    ─────1─2────────  (line l)
         3 4
         │
    ─────5─6────────  (line m, parallel to l)
         7 8
         │

Corresponding angles (equal): 1=5, 2=6, 3=7, 4=8
Alternate interior angles (equal): 3=6, 4=5
Co-interior/Same side (supplementary): 3+5=180°, 4+6=180°

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Chapter 6: The Triangle and Its Properties

6.1 Parts of a Triangle — Diagram

           A
          /\
         /  \
        /    \
  AB = c/      \b = AC
      /  angle A\
     /____________\
    B      a       C
          BC

a = side opposite to A
b = side opposite to B  
c = side opposite to C

6.2 Key Triangle Properties

ANGLE SUM PROPERTY:
∠A + ∠B + ∠C = 180°

Example: If ∠A = 60°, ∠B = 70°, find ∠C
60° + 70° + ∠C = 180°
∠C = 180° − 130° = 50°

EXTERIOR ANGLE PROPERTY:
Exterior angle = Sum of two non-adjacent interior angles

         A
        /│\
       / │ \
      /  │  \
     B───────C─────D
               ∠ACD (exterior)

∠ACD = ∠A + ∠B

6.3 Median and Altitude

MEDIAN: Vertex to MIDPOINT of opposite side

         A
        /|\
       / | \
      /  |  \
     B───M───C    M = midpoint of BC
      BM = MC

ALTITUDE: PERPENDICULAR from vertex to opposite side

         A
        /|
       / |
      /  | (height h)
     /   |
    B────D───C    AD ⊥ BC, ∠ADC = 90°

6.4 Pythagoras Theorem (Right Triangles)

         A
         │
         │  (height = b)
         │
         └──────────B
              (base = a)
     hypotenuse (c) = AB = longest side

FORMULA: c² = a² + b²

Example: a = 3, b = 4, find c
c² = 3² + 4² = 9 + 16 = 25
c = √25 = 5  ← (3-4-5 Pythagorean triple)

Common Pythagorean Triples:
  3, 4, 5    (3²+4²=5²)
  5, 12, 13  (5²+12²=13²)
  8, 15, 17  (8²+15²=17²)

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Chapter 7: Congruence of Triangles

7.1 What is Congruence?

Two figures are CONGRUENT if they have:
  ✓ Same shape
  ✓ Same size
  ✓ Perfectly overlap when placed on each other

   △ABC ≅ △DEF means:
   A↔D, B↔E, C↔F (vertices correspond)
   AB = DE, BC = EF, CA = FD (sides equal)
   ∠A = ∠D, ∠B = ∠E, ∠C = ∠F (angles equal)

7.2 Congruence Rules (SSS, SAS, ASA, RHS)

SSS (Side-Side-Side): All 3 sides equal
   a=d, b=e, c=f → △ABC ≅ △DEF

SAS (Side-Angle-Side): 2 sides + INCLUDED angle
   AB=DE, ∠B=∠E, BC=EF → △ABC ≅ △DEF
              ↑ angle between the two sides

ASA (Angle-Side-Angle): 2 angles + INCLUDED side
   ∠B=∠E, BC=EF, ∠C=∠F → △ABC ≅ △DEF

RHS (Right angle-Hypotenuse-Side):
   Right angle, hypotenuse equal, one side equal
   → Right triangles are congruent

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Chapter 8: Comparing Quantities

8.1 Ratio and Proportion

RATIO: Comparison of two quantities
   Ratio of 4 to 6 = 4:6 = 2:3 (simplify by HCF)

PROPORTION: Two ratios are equal
   4:6 = 2:3 → 4/6 = 2/3 → 4×3 = 6×2 → 12 = 12 ✓
   (Cross multiplication test)

UNITARY METHOD:
   If 5 pens cost ₹35, how much do 8 pens cost?
   1 pen = ₹35 ÷ 5 = ₹7
   8 pens = ₹7 × 8 = ₹56

8.2 Percentage — Diagram

"Per cent" = "per hundred"

   0%──────────────────────────100%
   │                            │
  None                         All

Converting:
  Fraction → %:  3/4 = (3÷4)×100 = 75%
  % → Fraction:  25% = 25/100 = 1/4
  % → Decimal:   45% = 45/100 = 0.45

8.3 Profit, Loss, Simple Interest

PROFIT/LOSS:
  Profit = SP − CP  (when SP > CP)
  Loss   = CP − SP  (when CP > SP)
  Profit% = (Profit/CP) × 100
  Loss%   = (Loss/CP) × 100

SIMPLE INTEREST:
  SI = (P × R × T) / 100
  
  P = Principal (original amount)
  R = Rate of interest per year (%)
  T = Time (years)
  
  Amount (A) = P + SI

Example: Find SI on ₹2000 at 5% for 3 years.

SI = (2000 × 5 × 3)/100 = 30000/100 = ₹300
Amount = 2000 + 300 = ₹2300

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Chapter 9: Rational Numbers

9.1 What are Rational Numbers?

RATIONAL NUMBERS = p/q form, where p,q are integers, q ≠ 0

Number Line for Rational Numbers:

  -2    -3/2   -1   -1/2    0   1/2    1   3/2    2
   ◄─────┼──────┼────┼──────┼───┼──────┼───┼──────►

Rational numbers include:
  ✓ Integers (3 = 3/1)
  ✓ Fractions (1/2, -3/4)
  ✓ Terminating decimals (0.25 = 1/4)
  ✓ Repeating decimals (0.333... = 1/3)

9.2 Properties

| Property | Addition | Multiplication | |----------|----------|---------------| | Closure | p/q + r/s = rational ✓ | p/q × r/s = rational ✓ | | Commutative | a+b = b+a ✓ | a×b = b×a ✓ | | Associative | (a+b)+c = a+(b+c) ✓ | (a×b)×c = a×(b×c) ✓ | | Identity | 0 (additive) | 1 (multiplicative) | | Inverse | a + (−a) = 0 | a × (1/a) = 1 |

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Chapter 10: Algebraic Expressions

10.1 Parts of an Expression — Diagram

  3x² − 5xy + 7y − 2
  ─┬─   ─┬─   ─┬   ─┬
   │      │     │    │
   │      │     │    └── Constant term (no variable)
   │      │     └─────── Term: 7y (coefficient=7, variable=y)
   │      └───────────── Term: −5xy (coefficient=−5, variables=x,y)
   └──────────────────── Term: 3x² (coefficient=3, variable=x, power=2)

Types of terms:
  Monomial: 1 term (e.g., 3xy)
  Binomial: 2 terms (e.g., x + 2)
  Trinomial: 3 terms (e.g., x² + 2x + 1)
  Polynomial: many terms

10.2 Like and Unlike Terms

LIKE TERMS: Same variable(s) with same powers
  3x and 7x are LIKE (both have x¹)
  4x²y and −2x²y are LIKE (both have x²y)
  
UNLIKE TERMS: Different variables or different powers
  3x and 3x² are UNLIKE (different powers)
  2xy and 2x are UNLIKE (different variables)

Adding LIKE TERMS:
  3x + 7x = (3+7)x = 10x
  5xy − 2xy = (5−2)xy = 3xy

10.3 Solving Example

Simplify: 3x² − 2x + 5 + 2x² + 4x − 3

Group like terms:
  (3x² + 2x²) + (−2x + 4x) + (5 − 3)
= 5x² + 2x + 2

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Chapter 11: Perimeter and Area

11.1 Formulas with Diagrams

TRIANGLE:
        /\
       /  \  h (height)
      /    \
     /______\
         b (base)

Area = (1/2) × base × height = (1/2)bh
Perimeter = sum of all 3 sides

PARALLELOGRAM:
  _________
 /         /
/___________/
b (base), h (height perpendicular to base)

Area = base × height = bh
Perimeter = 2(l + b)

CIRCLE:
       r
   ┌───┼───┐
   │   ·   │   r = radius
   └───────┘   d = diameter = 2r

Circumference = 2πr = πd
Area = πr²         (π ≈ 22/7 ≈ 3.14)

11.2 Example Problems

Find area of triangle: base = 12 cm, height = 8 cm

Area = (1/2) × 12 × 8 = 48 cm²

Find circumference and area of circle with r = 7 cm:

Circumference = 2 × (22/7) × 7 = 44 cm
Area = (22/7) × 7 × 7 = 154 cm²

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Quick Revision Checklist — Class 7 Maths

• [ ] Even negatives in product → positive; Odd negatives → negative
• [ ] Fraction division: multiply by reciprocal
• [ ] Mean = Sum/Count; Median = middle value; Mode = most frequent
• [ ] Simple equation: transpose with sign change
• [ ] Angle sum of triangle = 180°
• [ ] Exterior angle = sum of two interior opposite angles
• [ ] Pythagoras: c² = a² + b² for right triangles
• [ ] Congruence rules: SSS, SAS, ASA, RHS
• [ ] SI = PRT/100
• [ ] Profit% = (Profit/CP) × 100
• [ ] Area of circle = πr²; Area of triangle = ½ × b × h
• [ ] Like terms: same variable same power only